It was conjectured, twenty years ago, the following result that would generalize the so-called rank rigidity theorem for homogeneous Euclidean submanifolds: let M-n, n >= 2, be a full and irreducible homogeneous submanifold of the sphere SN-1 subset of R-N such that the normal holonomy group is not transitive (on the unit sphere of the normal space to the sphere). Then M-n must be an orbit of an irreducible s-representation (i.e. the isotropy representation of a semisimple Riemannian symmetric space).
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